Cause-Specific Cox or Fine-Gray: Two Hazards, Two Questions

อ่านฉบับภาษาไทย (Thai version)

Abstract

When patients can die before the event of interest, a treatment has two hazard ratios answering different questions. The cause-specific hazard ratio (csHR) asks how fast recurrence happens among patients still alive and recurrence-free; a csHR of 0.70 means a cause-specific recurrence hazard about 30% lower, not a 30% lower chance of recurrence. The Fine-Gray model targets the subdistribution hazard, whose risk set keeps patients who died, so its ratio (sHR) is tied to the cumulative incidence function (CIF) but is not an ordinary biological hazard ratio. Because the CIF of recurrence depends on both the recurrence and death hazards, a csHR alone cannot say how many patients recur. The CIF is a quantity; Fine-Gray is a regression model. An sHR of 0.70 means a lower subdistribution hazard, not a 30% lower risk, mortality or cause-specific hazard, and not an odds ratio. This article shows how to choose between models by the question asked, and why an sHR is never automatically converted into a percent risk reduction.

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A patient after cancer surgery can take two paths

An oncology team follows patients after surgery for cancer. Every patient starts in the same state: alive, with no recurrence. From there, one of two things can happen first: a recurrence, or death before any recurrence.

A patient who dies before recurrence can never have one afterwards. The probability of a later recurrence is 0. That is what makes death a competing risk for recurrence: an event that, once it happens, closes the door on the event of interest. It is not ordinary censoring, because a censored patient could still have a recurrence that nobody sees, while a patient who died can no longer have one.

The team's new treatment slows recurrence among patients who are still alive and recurrence-free. More treated patients, however, die before any recurrence. The team wants to know how the treatment changes the recurrence rate, and how many patients will actually have a recurrence. Those are two different questions, and they need two different hazard ratios.

All of this sits in the first-event, competing-risk world. Each patient has one first event, recurrence or death, and the tools are the Kaplan-Meier curve, the cumulative incidence function, cause-specific Cox and Fine-Gray. Repeated admissions in the same patient belong to a different world, the recurrent-event world of the Andersen-Gill model, covered in part 6, Recurrent Events: Counting Every Admission Without Pretending They Are Independent. Telling these two worlds apart is the first distinction to make.

One kind of event: 1 minus Kaplan-Meier is the risk

Start with the simplest case, death from any cause. There is no competing event, because every death counts as the same event. The Kaplan-Meier estimate (the usual step-curve estimate of the chance of staying event-free) gives $S(t) = P(T > t)$, the chance of not yet having had the event by time $t$.

If $S(5) = 0.80$, then 80% of patients have not had the event by 5 years. So $1 - S(5) = 0.20$: 20% have had it. With only one kind of event, one minus Kaplan-Meier is the cumulative probability of the event, and it works well.

Now let the event of interest be recurrence, $R$, and let patients also die before recurrence, $D$. The population at time $t$ splits into three parts that add up to 1:

$$1 = S(t) + \mathrm{CIF}_R(t) + \mathrm{CIF}_D(t)$$

Here $S(t)$ is the share still alive and recurrence-free, $\mathrm{CIF}_R(t)$ the share whose first event was a recurrence, and $\mathrm{CIF}_D(t)$ the share who died first.

Hand example: 100 patients at 5 years

Suppose 100 patients are followed for 5 years after cancer surgery, and every one of them is still under observation at 5 years.

1. Count the three outcomes

   \[ 50 + 20 + 30 = 100 \]

   50 patients are alive and recurrence-free, 20 have had a recurrence, and 30 died before any recurrence.

2. Write them as proportions

   \[ 1 = S(5) + \mathrm{CIF}_R(5) + \mathrm{CIF}_D(5) = 0.50 + 0.20 + 0.30 \]

   Every patient is in exactly one of the three groups, so the three proportions add up to 1.

3. One minus event-free survival

   \[ 1 - S(5) = 0.50 \]

   This is the probability of recurrence or death by 5 years, not the probability of recurrence alone.

4. The cumulative incidence of recurrence

   \[ \mathrm{CIF}_R(5) = 0.20 \]

   Of all patients at baseline, 20% have a recurrence within 5 years, allowing for those who die before recurrence.

Result: The real-world risk of recurrence by 5 years is 0.20. The 0.50 from one minus event-free survival counts recurrence or death, and only the cumulative incidence splits it into the two causes.

The CIF: how many patients really have a recurrence?

The cumulative incidence function (CIF) answers that question:

$$\mathrm{CIF}_R(t) = P(\text{recurrence by } t \text{, before a competing death})$$

$\mathrm{CIF}_R(5) = 0.20$ reads directly: of all patients at baseline, 20% have a recurrence within 5 years, allowing for the fact that some die before recurrence. It is a real-world absolute probability. It depends on every cause-specific hazard, not on the recurrence hazard alone [1].

With competing risks, the standard nonparametric estimate of the CIF is the Aalen-Johansen estimator, which treats a death as the end of any chance of recurrence. Naive one minus Kaplan-Meier, with deaths censored, treats a death as if the recurrence were merely unseen. In fact the recurrence is impossible. So with competing risks, $1 - \mathrm{KM}_R \ne \mathrm{CIF}_R$, and the naive figure usually overstates the risk.

Part 4, Why 1 Minus Kaplan-Meier Overstates Risk When Patients Can Die First, works through that count by hand.

Cause-specific Cox: how fast, among patients still event-free now?

The CIF asks how many patients have a recurrence. Sometimes the main interest is not a probability. The question is instead: among patients who are alive and recurrence-free now, how much does the treatment change the rate at which recurrence happens?

That is the cause-specific hazard of recurrence, $\lambda_R(t)$. A hazard is an instantaneous event rate among the patients still at risk at time $t$. It is not the probability that all patients will have a recurrence within 5 years.

A cause-specific Cox model relates this hazard to a covariate $X$, such as treatment:

$$\lambda_R(t \mid X) = \lambda_{R0}(t)\,e^{\beta X}$$

Here $\lambda_{R0}(t)$ is the baseline hazard, left unspecified, and the ratio between groups, $e^{\beta}$, is assumed constant over time (the proportional hazards assumption). That ratio is the cause-specific hazard ratio (csHR): an effect on the recurrence rate among patients who are still event-free [2].

How cause-specific Cox handles a death

With recurrence as the event of interest, a recurrence is the event. A death before recurrence takes the patient out of the recurrence risk set, the patients counted as at risk, at the time of death. When the Cox model is fitted, that death is therefore coded like a censoring at that time.

But beware: this is correct for estimating the cause-specific hazard. It does not mean that one minus Kaplan-Meier with deaths censored can be called the probability of recurrence. This is the point where readers are most often misled.

Reading a csHR of 0.70

Suppose a cause-specific Cox model for recurrence gives a csHR of 0.70 for treatment versus control.

It means that at any given time, among patients still alive and recurrence-free, the treated group has a cause-specific recurrence hazard about 30% lower than the control group. That reading relies on the proportional hazards assumption.

It does not mean that the treatment lowers the probability of recurrence by 30%. The actual probability of recurrence also depends on death.

Why Fine-Gray exists

Now suppose the clinical question is not how fast recurrence happens among patients still at risk. It is how the treatment is associated with the probability that a patient has a recurrence in the real world, when death competes. That is the world of the Fine-Gray model.

Fine-Gray does not model the ordinary cause-specific hazard. It models the subdistribution hazard, which is tied more directly to the CIF. Its output is the subdistribution hazard ratio (sHR). The model was designed to study how covariates relate to the cumulative incidence when competing risks are present [3].

A typical output is an sHR of 0.70 for treatment versus control. Its direct reading is narrow.

The odd thing Fine-Gray does

In cause-specific Cox, a patient who dies leaves the recurrence risk set. That makes sense, because the patient can no longer have a recurrence.

Fine-Gray builds a special mathematical risk set that still keeps the patients who had the competing event. It does this so that the subdistribution hazard connects to the CIF [4]. This is why the model confuses so many readers.

So do not read a Fine-Gray sHR the way you read a conventional hazard ratio. The small example below puts the two risk sets side by side.

Hand example: two risk sets at month 6

Twelve invented patients are followed for recurrence after cancer surgery, and death before recurrence is the competing event. Their follow-up, in months after surgery:

• deaths at 2, 5 and 9;
• recurrences at 3, 6, 7 and 10;
• censored at 4, 8, 11, 12 and 12.

A recurrence happens at month 6, so build both risk sets at that time.

1. Cause-specific risk set

   \[ n_{\mathrm{cs}}(6) = 8 \]

   Eight patients are still alive, recurrence-free and under observation just before month 6. The deaths at months 2 and 5 have left this set.

2. Fine-Gray risk set

   \[ n_{\mathrm{sd}}(6) = 8 + 2 = 10 \]

   The two patients who died at months 2 and 5 are kept in, although they can never have a recurrence. The patient censored at month 4 and the recurrence at month 3 are in neither set.

3. Cause-specific hazard at month 6

   \[ \hat\lambda_R(6) = \frac{1}{8} = 0.125 \]

   One recurrence over the 8 patients who can still have one. A hat marks an estimate from data.

4. Subdistribution hazard at month 6

   \[ \hat h_R(6) = \frac{1}{10} = 0.10 \]

   The same recurrence over the larger Fine-Gray risk set.

Result: One recurrence, two denominators: 8 in the cause-specific risk set and 10 in the Fine-Gray risk set. The real Fine-Gray estimator does not count each kept death in full; it down-weights it by the chance that patient would still have been under observation, so the weighted denominator is at most 10. The patients who died stay in the Fine-Gray set, so its hazard is not a rate that any living patient experiences.

Cause-specific Cox versus Fine-Gray

The example that separates them most clearly

Suppose a drug lowers the recurrence hazard, $\lambda_R$, but raises the death hazard, $\lambda_D$, a lot. This is the treatment from the opening scene.

A cause-specific Cox model for recurrence may give $\mathrm{csHR}_R = 0.70$. Among patients still alive and recurrence-free, the recurrence process is slower. The clinical reality, though, is that the treatment also kills more patients.

So the actual probability of recurrence, $\mathrm{CIF}_R(t)$, is not determined by 0.70 alone; the death hazard has to be taken into account. The CIF does not have to fall by the same 30%. Nor must it rise because more patients die: more deaths leave fewer patients able to have a recurrence. The net result depends on both hazards.

Why the CIF needs both hazards

The cause-specific recurrence hazard, $\lambda_R(t)$, and the cause-specific death hazard, $\lambda_D(t)$, together build the overall event-free survival, $S(t)$. Both recurrence and death take patients out of the event-free state. Then:

$$\mathrm{CIF}_R(t) = \int_0^t S(u^-)\,\lambda_R(u)\,du$$

In words, the probability of recurrence adds up, over follow-up, the chance of still being alive and event-free just before each moment, $S(u^-)$, times the speed of recurrence at that moment, $\lambda_R(u)$. Because $S$ falls when either hazard rises, the risk of recurrence is a function of both:

$$\mathrm{CIF}_R = f(\lambda_R, \lambda_D)$$

That is why a cause-specific Cox model for recurrence alone cannot give the full recurrence CIF; the competing hazards are needed too. With competing events, the one-to-one link between a hazard and a risk is lost [5].

Choose by the question: probability or hazard?

Once a competing event is present, the first thing to settle is whether the question asks about a probability or about a hazard.

To describe a probability, such as the share of patients with a recurrence at 5 years when some die first, estimate the CIF, not one minus Kaplan-Meier. To run a regression on the event rate among patients currently event-free, use cause-specific Cox. To run a regression on the real-world cumulative incidence, use Fine-Gray.

Methodological work recommends choosing by the research question [4], not by a rule that a competing risk always calls for Fine-Gray. The choice should follow the scientific estimand, not whichever model is more familiar.

Which question, which approach

Words that point to the method

The CIF is not Fine-Gray

Many readers merge these two, and they should not. The CIF is a quantity, a function of time: $\mathrm{CIF}_R(t) = P(\text{recurrence by } t)$. It is the thing we want to estimate.

Fine-Gray is a regression model. It studies the effect of a covariate $X$ on the subdistribution hazard, which is linked to the CIF.

A useful analogy is $\mathrm{KM} : \mathrm{Cox} \approx \mathrm{CIF} : \text{Fine-Gray}$. Kaplan-Meier and the CIF give descriptive curves and probabilities; Cox and Fine-Gray are regressions with covariates.

Not rivals: two models, two questions

Cause-specific Cox and Fine-Gray are not competitors where only one may ever be chosen. They answer different questions.

A treatment may have $\mathrm{csHR}_R = 0.70$ while its $\mathrm{sHR}_R$ is not 0.70, because the CIF, and so Fine-Gray, is also moved by competing death. The reverse holds too: a Fine-Gray sHR of 0.70 does not imply that the cause-specific recurrence hazard ratio is also 0.70.

In the drug example, the cause-specific Cox model for recurrence shows only the first part, the lower recurrence hazard. The CIF of recurrence comes from $\lambda_R$ and $\lambda_D$ together. That is why a competing-risk analysis is best reported with every cause-specific hazard and every cumulative incidence function [6].

Four ways to misread an sHR of 0.70

• "The recurrence probability is exactly 30% lower at every time."

  An sHR is not a risk ratio. Published Fine-Gray analyses often slip into this reading [2].

  Fix: Say that the recurrence subdistribution hazard is lower, and read the risk itself, at a stated time, from the cumulative incidence curves.

• "The cause-specific recurrence hazard is 30% lower."

  That is a different estimand. The cause-specific hazard is built on a different risk set, and its ratio need not be 0.70.

  Fix: If the question is the recurrence rate among patients still event-free, fit a cause-specific Cox model and report its csHR.

• "The mortality hazard is reduced by 30%."

  The modelled event was recurrence, not death.

  Fix: To say anything about death, model death itself, with a cause-specific Cox model or a CIF for death before recurrence.

• "The odds of recurrence-free survival are 0.70."

  A hazard is not an odds.

  Fix: Keep the words subdistribution hazard ratio, and never turn the sHR into an odds ratio.

Five sentences to remember

The final decision picture

For any time-to-event question, walk down this path.

1. Does the event repeat in the same patient, and should every occurrence count? If yes, use Andersen-Gill (see part 6, Recurrent Events: Counting Every Admission Without Pretending They Are Independent).
2. If not, is there a competing event that stops the event of interest from ever happening? If not, Kaplan-Meier and Cox are enough.
3. If there is a competing event, match the method to the question:
   • the actual probability of the event: the CIF;
   • that probability related to covariates: Fine-Gray;
   • the instantaneous cause-specific rate related to covariates: cause-specific Cox.

Where does Kaplan-Meier sit? It is the base when the outcome is a single terminal event, or a composite in which every event counts as failure. Overall survival, death from any cause, has no competing cause, so Kaplan-Meier fits it well: $S(t) = P(\text{alive at } t)$.

Split death into cardiovascular and non-cardiovascular death, though, and non-cardiovascular death becomes a competing risk for cardiovascular death. The actual probability of cardiovascular death is then $\mathrm{CIF}_{CV}$, not $1 - \mathrm{KM}_{CV}$ with the non-cardiovascular deaths censored.

Glossary

competing risk

An event that, once it happens, prevents the event of interest from ever happening afterwards, such as death before recurrence.

cumulative incidence function (อุบัติการณ์สะสม (cumulative incidence function, CIF))

The probability of having had a given event by time t, allowing for patients who have a competing event first; a quantity, not a model.

cause-specific hazard

The instantaneous rate of one cause among patients still free of any event; a patient who has a competing event leaves its risk set.

cause-specific hazard ratio (อัตราส่วนฮาซาร์ดเฉพาะสาเหตุ)

The ratio between groups of the cause-specific hazard; an effect on the event rate among patients still event-free, not on the probability of the event.

subdistribution hazard

The hazard of one cause over a special risk set that also keeps patients who already had a competing event, built so that it links to the cumulative incidence of that cause.

subdistribution hazard ratio (อัตราส่วนฮาซาร์ดของ subdistribution)

The ratio between groups of the subdistribution hazard; not a risk ratio, not an odds ratio and not an ordinary biological hazard ratio.

risk set

The patients counted as at risk at a given time, the denominator of a hazard.

Fine-Gray model (แบบจำลอง Fine-Gray)

A proportional hazards regression for the subdistribution hazard of one cause, used for questions about cumulative incidence.

estimand

The quantity an analysis sets out to estimate, such as a cause-specific hazard ratio or a cumulative incidence at a stated time.

aetiological versus prognostic question

An aetiological question asks how fast an event happens among patients still at risk; a prognostic question asks how likely the event is by a given time.

References

1. Putter H, Fiocco M, Geskus RB. Tutorial in biostatistics: competing risks and multi-state models. Stat Med. 2007;26(11):2389-2430. doi:10.1002/sim.2712 https://doi.org/10.1002/sim.2712
2. Austin PC, Fine JP. Practical recommendations for reporting Fine-Gray model analyses for competing risk data. Stat Med. 2017;36(27):4391-4400. doi:10.1002/sim.7501 https://doi.org/10.1002/sim.7501
3. Fine JP, Gray RJ. A proportional hazards model for the subdistribution of a competing risk. J Am Stat Assoc. 1999;94(446):496-509. doi:10.1080/01621459.1999.10474144 https://doi.org/10.1080/01621459.1999.10474144
4. Lau B, Cole SR, Gange SJ. Competing risk regression models for epidemiologic data. Am J Epidemiol. 2009;170(2):244-256. doi:10.1093/aje/kwp107 https://doi.org/10.1093/aje/kwp107
5. Andersen PK, Geskus RB, de Witte T, Putter H. Competing risks in epidemiology: possibilities and pitfalls. Int J Epidemiol. 2012;41(3):861-870. doi:10.1093/ije/dyr213 https://doi.org/10.1093/ije/dyr213
6. Latouche A, Allignol A, Beyersmann J, Labopin M, Fine JP. A competing risks analysis should report results on all cause-specific hazards and cumulative incidence functions. J Clin Epidemiol. 2013;66(6):648-653. doi:10.1016/j.jclinepi.2012.09.017 https://doi.org/10.1016/j.jclinepi.2012.09.017

Key takeaways

• A cause-specific hazard ratio answers how fast an event happens among patients still alive and event-free; a csHR of 0.70 is a hazard about 30% lower, not a 30% lower probability of the event.
• Fine-Gray models the subdistribution hazard, whose risk set keeps patients who had the competing event, so its sHR is tied to the cumulative incidence and is not an ordinary biological hazard ratio.
• The cumulative incidence of recurrence depends on the recurrence hazard and the death hazard together, so a drug that lowers recurrence but raises death cannot be judged by its csHR alone.
• The CIF is a quantity and Fine-Gray is a regression model; cause-specific Cox and Fine-Gray are not rivals, and the choice follows the question, probability or hazard.
• Fine-Gray models the subdistribution hazard; never automatically convert an sHR into a percent risk reduction.

Related in the wiki: [[cumulative-incidence-competing-risks-kaplan-meier]] [[time-to-event-survival-analysis]]